Customer Churn using Survival Analysis (Part 1)

• 1 Business Understanding
  • 1.1 Modeling customer churn
  • 1.2 Survival and Censoring Times
  • 1.3 Other survival analysis terminology
  • 1.4 The goal of survival analysis
• 2 Data Understanding
  • response variables
• 3 Data Preparation
  • 3.1 Event/Censor times/durations
  • 3.2 Status indicator
  • 3.3 Numericals
  • 3.4 Categoricals
• 4 Modeling
  • 4.1 Survival Probability for the average customer
  • 4.2 Survival Probability for the average customer based on segments
  • 4.3 Estimate Hazard
  • 4.4 Nelson-Aalen estimator
    • 4.4.1 Cumulative hazard function
    • 4.4.2 Hazard function
  • 4.2 Survival Regression Model
    • 4.2.1 Proportional Hazards Assumption
    • 4.2.2 Cox Proportional Hazards Regression
    • 4.2.3 Split the data
• 5 Evaluation
  • 5.1 Plot the coefficients
  • 5.2 Predict on the train set
  • 5.3 Predict on the validation set
  • 5.4 Concordance
  • 5.5 Check assumptions

1 Business Understanding

Customer churn is a term that refers to the event that an existing customer (or client) of a company decides to separate from the company. This could mean that the customer switches to buying from a competitor, or simply, that the customer stops doing business with the company.

1.1 Modeling customer churn

It is important for digitally enabled companies to track customer churn. To model this aspect of customer behavior the first approach that comes to mind is supervised learning. A dataset will contain some features of customers and a target variable: Whether a customer has churned or not. A classification model could be helpful to identify customers that might churn some or other time. But will this be useful? Eventually, all customers will churn at some future time. What might be much more useful is to predict the lifetime of a customer, in other words, how long a customer will stay engaged with the company.

When predicting the lifetime of a customer, we change from a classification to a regression model as we need to predict a numerical value (lifetime). This regression approach, however, has a serious drawback. Most customers will not have churned, at a specific point in time. This means for all these datapoints we do not have a target value. We can choose to eliminate these datapoints from the dataset, but they represent the larger part of it! In addition, the fact that all these customers are still engaged with the company represent valuable information. It will be beneficial to have a method that can handle this complication. Such a method is that of Survival Analysis.

Survival analysis focuses on a unique kind of outcome variable: the time until an event occurs. In our present situation this event is the 'death' or churn of a customer. This will allow us to predict the time until a specific customer churns. For all customers in the dataset who are still active, we simply do not know when they will churn, and we simply flag their time-to-churn as censored. A customer record may also be censored in other cases, e.g. when a customer's data are lost, or when a customer engaged with the company earlier than the date range of the dataset.

1.2 Survival and Censoring Times

We assume that for each customer, there is a

• Survival time, $T$

This is the time at which the event of interest occurs. Survival time is also referred to as event time or failure time. In our present case this refers to the duration from engagement with the company until the cancellation of subscription, say (i.e. churn.

• Censoring time $C$

This is time at which censoring occurs. This could be the time at which the last assessment of the cusomer is undertaken, or the time at which the dataset is compiled (given the customer is still engaged).

In our dataset, we will have two outcome variables:

• Duration $Y$

We observe a duration which could be either a survival time $T$, or else a censoring time $C$. In fact, we observe the random variable $Y = min(T, C)$. If $T<C$ we observe the survival time $T$. Alternatively, if $T>C$ we observe the censoring time $C$.

• Event/Status indicator $\delta$

In addition to observing a duration, we also observe a status or event indicator,

$$ \delta = \begin{cases} 1 \text{ if } T \leq C\\ 0 \text{ if } T > C \end{cases} $$

When we observe a survival time $\delta=1$, and if we instead observe a censoring time $\delta=0$.

• Dataset

The dataset consists of a set of datapoints $(x_1, y_1, \delta_1), ..., (x_i, y_i, \delta_i), ..., (x_n, y_n, \delta_n)$ where

• $x_i$ is the feature vector
• $y_i$ is the duration, either a survival time or a censoring time
• $\delta_i$ is the status/event indicator

This means we actually have two outcome variables: $y$ and $\delta$. The $x$'s are the usual predictor points.

1.3 Other survival analysis terminology

• Usual statistical functions

The survival time $T$ is a random variable and the usual statistical properties apply. First there is the cumulative distribution function (cdf) of $T$:

$F(t) = Pr\{T \leq t\}$

Next, there is the probability density function (pdf) of $T$:

$f(t)$

which is also known as the event density function, i.e. the rate of events per unit time.

The usual relationships exist between $F(t)$ and $f(t)$:

$F(t) = \int_{-\infty}^{t}f(u)du$

$f(t) = \frac{dF(t)}{dt}$

• Survival function/probability

$S(t) = Pr\{T > t\}$

Survival probability in $t$ is the probability that the survival time, $T$ exceeds $t$, i.e. the probability to survive beyond $t$.

Note that

$1 - F(t) = S(t)$, and

$f(t) = -\frac{dS(t)}{dt}$

• Survival function estimate

When we estimate the survival probability with numerical techniques, we usually refer to the survival function estimate

$\hat{S}(t)$

• Hazard function

The hazard function expresses the conditional probability that the event will occur within $t, t+dt$, given that it has not occurred before

$h(t) = \lim_{\Delta t \to 0} \frac{Pr\{t < T \leq t+\Delta t | T > t \}}{\Delta t} = \frac{f(t)}{S(t)} = -\frac{d}{dt}S(t)$

$h(t)$ is the event rate in the instant after time $t$, given survival past that time.

• Cumulative hazard function

$H(t) = \int_{0}^{t}h(u)du$

• Some relationships

$S(t) = \text{exp}[-\int_{0}^{t}h(u)du]$

$S(t) = \text{exp}[-H(t)]$

$h(t) = -[\frac{\frac{dS(t)}{dt}}{S(t)}]$

1.4 The goal of survival analysis

The goal of survival analysis may be simply stated as to:

Estimate the time to the event of interest, $T_j$ (survival time), for a new instance $j$ with feature predictors denoted by $X_j$.

2 Data Understanding

!pip install lifelines

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import warnings
warnings.filterwarnings('ignore') #6
%matplotlib inline
%config InlineBackend.figure_format = 'retina'
from matplotlib import pyplot as plt
import seaborn as sns #7
import pandas as pd
pd.set_option('display.max_columns', None)
import lifelines

print('pandas:', pd.__version__)
print('lifelines:', lifelines.__version__)

pandas: 1.3.5
lifelines: 0.27.0

Pointers related to the dataset:

https://github.com/treselle-systems/customer_churn_analysis https://www.kaggle.com/datasets/blastchar/telco-customer-churn

Other customer churn datasets:

https://community.ibm.com/community/user/businessanalytics/blogs/steven-macko/2019/07/11/telco-customer-churn-1113

df = pd.read_csv('https://raw.githubusercontent.com/'
                         'treselle-systems/customer_churn_analysis/'
                         'master/WA_Fn-UseC_-Telco-Customer-Churn.csv')
# df.head(10)
df

	customerID	gender	SeniorCitizen	Partner	Dependents	tenure	PhoneService	MultipleLines	InternetService	OnlineSecurity	OnlineBackup	DeviceProtection	TechSupport	StreamingTV	StreamingMovies	Contract	PaperlessBilling	PaymentMethod	MonthlyCharges	TotalCharges	Churn
0	7590-VHVEG	Female	0	Yes	No	1	No	No phone service	DSL	No	Yes	No	No	No	No	Month-to-month	Yes	Electronic check	29.85	29.85	No
1	5575-GNVDE	Male	0	No	No	34	Yes	No	DSL	Yes	No	Yes	No	No	No	One year	No	Mailed check	56.95	1889.5	No
2	3668-QPYBK	Male	0	No	No	2	Yes	No	DSL	Yes	Yes	No	No	No	No	Month-to-month	Yes	Mailed check	53.85	108.15	Yes
3	7795-CFOCW	Male	0	No	No	45	No	No phone service	DSL	Yes	No	Yes	Yes	No	No	One year	No	Bank transfer (automatic)	42.30	1840.75	No
4	9237-HQITU	Female	0	No	No	2	Yes	No	Fiber optic	No	No	No	No	No	No	Month-to-month	Yes	Electronic check	70.70	151.65	Yes
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
7038	6840-RESVB	Male	0	Yes	Yes	24	Yes	Yes	DSL	Yes	No	Yes	Yes	Yes	Yes	One year	Yes	Mailed check	84.80	1990.5	No
7039	2234-XADUH	Female	0	Yes	Yes	72	Yes	Yes	Fiber optic	No	Yes	Yes	No	Yes	Yes	One year	Yes	Credit card (automatic)	103.20	7362.9	No
7040	4801-JZAZL	Female	0	Yes	Yes	11	No	No phone service	DSL	Yes	No	No	No	No	No	Month-to-month	Yes	Electronic check	29.60	346.45	No
7041	8361-LTMKD	Male	1	Yes	No	4	Yes	Yes	Fiber optic	No	No	No	No	No	No	Month-to-month	Yes	Mailed check	74.40	306.6	Yes
7042	3186-AJIEK	Male	0	No	No	66	Yes	No	Fiber optic	Yes	No	Yes	Yes	Yes	Yes	Two year	Yes	Bank transfer (automatic)	105.65	6844.5	No

7043 rows × 21 columns

df.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 7043 entries, 0 to 7042
Data columns (total 21 columns):
 #   Column            Non-Null Count  Dtype  
---  ------            --------------  -----  
 0   customerID        7043 non-null   object 
 1   gender            7043 non-null   object 
 2   SeniorCitizen     7043 non-null   int64  
 3   Partner           7043 non-null   object 
 4   Dependents        7043 non-null   object 
 5   tenure            7043 non-null   int64  
 6   PhoneService      7043 non-null   object 
 7   MultipleLines     7043 non-null   object 
 8   InternetService   7043 non-null   object 
 9   OnlineSecurity    7043 non-null   object 
 10  OnlineBackup      7043 non-null   object 
 11  DeviceProtection  7043 non-null   object 
 12  TechSupport       7043 non-null   object 
 13  StreamingTV       7043 non-null   object 
 14  StreamingMovies   7043 non-null   object 
 15  Contract          7043 non-null   object 
 16  PaperlessBilling  7043 non-null   object 
 17  PaymentMethod     7043 non-null   object 
 18  MonthlyCharges    7043 non-null   float64
 19  TotalCharges      7043 non-null   object 
 20  Churn             7043 non-null   object 
dtypes: float64(1), int64(2), object(18)
memory usage: 1.1+ MB

# need to fix TotalCharges for better understanding of this feature
df[df['TotalCharges']==' ']

	customerID	gender	SeniorCitizen	Partner	Dependents	tenure	PhoneService	MultipleLines	InternetService	OnlineSecurity	OnlineBackup	DeviceProtection	TechSupport	StreamingTV	StreamingMovies	Contract	PaperlessBilling	PaymentMethod	MonthlyCharges	TotalCharges	Churn
488	4472-LVYGI	Female	0	Yes	Yes	0	No	No phone service	DSL	Yes	No	Yes	Yes	Yes	No	Two year	Yes	Bank transfer (automatic)	52.55		No
753	3115-CZMZD	Male	0	No	Yes	0	Yes	No	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	Two year	No	Mailed check	20.25		No
936	5709-LVOEQ	Female	0	Yes	Yes	0	Yes	No	DSL	Yes	Yes	Yes	No	Yes	Yes	Two year	No	Mailed check	80.85		No
1082	4367-NUYAO	Male	0	Yes	Yes	0	Yes	Yes	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	Two year	No	Mailed check	25.75		No
1340	1371-DWPAZ	Female	0	Yes	Yes	0	No	No phone service	DSL	Yes	Yes	Yes	Yes	Yes	No	Two year	No	Credit card (automatic)	56.05		No
3331	7644-OMVMY	Male	0	Yes	Yes	0	Yes	No	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	Two year	No	Mailed check	19.85		No
3826	3213-VVOLG	Male	0	Yes	Yes	0	Yes	Yes	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	Two year	No	Mailed check	25.35		No
4380	2520-SGTTA	Female	0	Yes	Yes	0	Yes	No	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	Two year	No	Mailed check	20.00		No
5218	2923-ARZLG	Male	0	Yes	Yes	0	Yes	No	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	One year	Yes	Mailed check	19.70		No
6670	4075-WKNIU	Female	0	Yes	Yes	0	Yes	Yes	DSL	No	Yes	Yes	Yes	Yes	No	Two year	No	Mailed check	73.35		No
6754	2775-SEFEE	Male	0	No	Yes	0	Yes	Yes	DSL	Yes	Yes	No	Yes	No	No	Two year	Yes	Bank transfer (automatic)	61.90		No

df['TotalCharges'] = pd.to_numeric(df['TotalCharges'], errors='coerce')
df

	customerID	gender	SeniorCitizen	Partner	Dependents	tenure	PhoneService	MultipleLines	InternetService	OnlineSecurity	OnlineBackup	DeviceProtection	TechSupport	StreamingTV	StreamingMovies	Contract	PaperlessBilling	PaymentMethod	MonthlyCharges	TotalCharges	Churn
0	7590-VHVEG	Female	0	Yes	No	1	No	No phone service	DSL	No	Yes	No	No	No	No	Month-to-month	Yes	Electronic check	29.85	29.85	No
1	5575-GNVDE	Male	0	No	No	34	Yes	No	DSL	Yes	No	Yes	No	No	No	One year	No	Mailed check	56.95	1889.50	No
2	3668-QPYBK	Male	0	No	No	2	Yes	No	DSL	Yes	Yes	No	No	No	No	Month-to-month	Yes	Mailed check	53.85	108.15	Yes
3	7795-CFOCW	Male	0	No	No	45	No	No phone service	DSL	Yes	No	Yes	Yes	No	No	One year	No	Bank transfer (automatic)	42.30	1840.75	No
4	9237-HQITU	Female	0	No	No	2	Yes	No	Fiber optic	No	No	No	No	No	No	Month-to-month	Yes	Electronic check	70.70	151.65	Yes
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
7038	6840-RESVB	Male	0	Yes	Yes	24	Yes	Yes	DSL	Yes	No	Yes	Yes	Yes	Yes	One year	Yes	Mailed check	84.80	1990.50	No
7039	2234-XADUH	Female	0	Yes	Yes	72	Yes	Yes	Fiber optic	No	Yes	Yes	No	Yes	Yes	One year	Yes	Credit card (automatic)	103.20	7362.90	No
7040	4801-JZAZL	Female	0	Yes	Yes	11	No	No phone service	DSL	Yes	No	No	No	No	No	Month-to-month	Yes	Electronic check	29.60	346.45	No
7041	8361-LTMKD	Male	1	Yes	No	4	Yes	Yes	Fiber optic	No	No	No	No	No	No	Month-to-month	Yes	Mailed check	74.40	306.60	Yes
7042	3186-AJIEK	Male	0	No	No	66	Yes	No	Fiber optic	Yes	No	Yes	Yes	Yes	Yes	Two year	Yes	Bank transfer (automatic)	105.65	6844.50	No

7043 rows × 21 columns

df['TotalCharges'].isnull().sum()

11

df_desc = df.describe(include="all").T
df_desc

	count	unique	top	freq	mean	std	min	25%	50%	75%	max
customerID	7043	7043	7590-VHVEG	1	NaN	NaN	NaN	NaN	NaN	NaN	NaN
gender	7043	2	Male	3555	NaN	NaN	NaN	NaN	NaN	NaN	NaN
SeniorCitizen	7043.0	NaN	NaN	NaN	0.162147	0.368612	0.0	0.0	0.0	0.0	1.0
Partner	7043	2	No	3641	NaN	NaN	NaN	NaN	NaN	NaN	NaN
Dependents	7043	2	No	4933	NaN	NaN	NaN	NaN	NaN	NaN	NaN
tenure	7043.0	NaN	NaN	NaN	32.371149	24.559481	0.0	9.0	29.0	55.0	72.0
PhoneService	7043	2	Yes	6361	NaN	NaN	NaN	NaN	NaN	NaN	NaN
MultipleLines	7043	3	No	3390	NaN	NaN	NaN	NaN	NaN	NaN	NaN
InternetService	7043	3	Fiber optic	3096	NaN	NaN	NaN	NaN	NaN	NaN	NaN
OnlineSecurity	7043	3	No	3498	NaN	NaN	NaN	NaN	NaN	NaN	NaN
OnlineBackup	7043	3	No	3088	NaN	NaN	NaN	NaN	NaN	NaN	NaN
DeviceProtection	7043	3	No	3095	NaN	NaN	NaN	NaN	NaN	NaN	NaN
TechSupport	7043	3	No	3473	NaN	NaN	NaN	NaN	NaN	NaN	NaN
StreamingTV	7043	3	No	2810	NaN	NaN	NaN	NaN	NaN	NaN	NaN
StreamingMovies	7043	3	No	2785	NaN	NaN	NaN	NaN	NaN	NaN	NaN
Contract	7043	3	Month-to-month	3875	NaN	NaN	NaN	NaN	NaN	NaN	NaN
PaperlessBilling	7043	2	Yes	4171	NaN	NaN	NaN	NaN	NaN	NaN	NaN
PaymentMethod	7043	4	Electronic check	2365	NaN	NaN	NaN	NaN	NaN	NaN	NaN
MonthlyCharges	7043.0	NaN	NaN	NaN	64.761692	30.090047	18.25	35.5	70.35	89.85	118.75
TotalCharges	7032.0	NaN	NaN	NaN	2283.300441	2266.771362	18.8	401.45	1397.475	3794.7375	8684.8
Churn	7043	2	No	5174	NaN	NaN	NaN	NaN	NaN	NaN	NaN

df.nunique()

customerID          7043
gender                 2
SeniorCitizen          2
Partner                2
Dependents             2
tenure                73
PhoneService           2
MultipleLines          3
InternetService        3
OnlineSecurity         3
OnlineBackup           3
DeviceProtection       3
TechSupport            3
StreamingTV            3
StreamingMovies        3
Contract               3
PaperlessBilling       2
PaymentMethod          4
MonthlyCharges      1585
TotalCharges        6530
Churn                  2
dtype: int64

[print(f"{col}: {df[col].unique()}") for col in df]

customerID: ['7590-VHVEG' '5575-GNVDE' '3668-QPYBK' ... '4801-JZAZL' '8361-LTMKD'
 '3186-AJIEK']
gender: ['Female' 'Male']
SeniorCitizen: [0 1]
Partner: ['Yes' 'No']
Dependents: ['No' 'Yes']
tenure: [ 1 34  2 45  8 22 10 28 62 13 16 58 49 25 69 52 71 21 12 30 47 72 17 27
  5 46 11 70 63 43 15 60 18 66  9  3 31 50 64 56  7 42 35 48 29 65 38 68
 32 55 37 36 41  6  4 33 67 23 57 61 14 20 53 40 59 24 44 19 54 51 26  0
 39]
PhoneService: ['No' 'Yes']
MultipleLines: ['No phone service' 'No' 'Yes']
InternetService: ['DSL' 'Fiber optic' 'No']
OnlineSecurity: ['No' 'Yes' 'No internet service']
OnlineBackup: ['Yes' 'No' 'No internet service']
DeviceProtection: ['No' 'Yes' 'No internet service']
TechSupport: ['No' 'Yes' 'No internet service']
StreamingTV: ['No' 'Yes' 'No internet service']
StreamingMovies: ['No' 'Yes' 'No internet service']
Contract: ['Month-to-month' 'One year' 'Two year']
PaperlessBilling: ['Yes' 'No']
PaymentMethod: ['Electronic check' 'Mailed check' 'Bank transfer (automatic)'
 'Credit card (automatic)']
MonthlyCharges: [29.85 56.95 53.85 ... 63.1  44.2  78.7 ]
TotalCharges: [  29.85 1889.5   108.15 ...  346.45  306.6  6844.5 ]
Churn: ['No' 'Yes']

[None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None]

# check missing values
df.isnull().sum()

customerID           0
gender               0
SeniorCitizen        0
Partner              0
Dependents           0
tenure               0
PhoneService         0
MultipleLines        0
InternetService      0
OnlineSecurity       0
OnlineBackup         0
DeviceProtection     0
TechSupport          0
StreamingTV          0
StreamingMovies      0
Contract             0
PaperlessBilling     0
PaymentMethod        0
MonthlyCharges       0
TotalCharges        11
Churn                0
dtype: int64

response variables

The role of response variables is played by 'tenure' for $y$ and 'Churn' for $\delta$.

plt.style.use('fivethirtyeight') #6
plt.rcParams['figure.figsize'] = [14, 8] #6
plt.rcParams['lines.linewidth'] = 2.5 #6

df['tenure'].hist(color='skyblue');

df['Churn'].value_counts(normalize=True).plot.bar(color=['lightgreen','red']);

df['MonthlyCharges'].hist(color='skyblue');

df['TotalCharges'].hist(color='skyblue');

Mth = sns.kdeplot(df['MonthlyCharges'][(df["Churn"]=='No')],
                color="Red", shade = True)
Mth = sns.kdeplot(df['MonthlyCharges'][(df["Churn"]=='Yes')],
                ax=Mth, color="Blue", shade= True)
Mth.legend(["No Churn", "Churn"], loc='upper right')
Mth.set_ylabel('Density')
Mth.set_xlabel('Monthly Charges')
Mth.set_title('Monthly charges by churn');

mypal = [
    'lightblue',    
    'lightpink',
    'lightcyan',
    'lightgreen',
    'khaki',
    'gold',
    'wheat',
    'peachpuff',
    'mistyrose',
    'sandybrown',
    'violet',
]

pd.crosstab(df['gender'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['SeniorCitizen'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['Partner'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['Dependents'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['Contract'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['PaperlessBilling'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['PaymentMethod'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['PhoneService'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['MultipleLines'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['InternetService'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['OnlineSecurity'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['OnlineBackup'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['DeviceProtection'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['TechSupport'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['StreamingTV'], df['Churn']).plot.bar(color=mypal, stacked=True);

pd.crosstab(df['StreamingMovies'], df['Churn']).plot.bar(color=mypal, stacked=True);

import numpy as np
# sns.heatmap(df.corr(), cmap="Paired")
# Generate a mask for the upper triangle #7
corr = df.corr()
mask = np.triu(np.ones_like(corr, dtype=np.bool))
# Generate a custom diverging colormap
cmap = sns.diverging_palette(220, 10, as_cmap=True)
# Plot correlation matrix
sns.heatmap(corr, cmap=cmap, linewidths=.5, mask=mask);

# inspect life times of a sample of customers
df_life = df[['tenure','Churn']]
df_life = df_life.replace({'Churn': {'No': False, 'Yes': True}})
df_life

	tenure	Churn
0	1	False
1	34	False
2	2	True
3	45	False
4	2	True
...	...	...
7038	24	False
7039	72	False
7040	11	False
7041	4	True
7042	66	False

7043 rows × 2 columns

from lifelines.plotting import plot_lifetimes
n = 40
sample = df_life.sample(n, replace=False)
durations = sample['tenure']
status = sample['Churn']

plt.figure(figsize=(18, 15));
plot_lifetimes(
    durations=durations, event_observed=status,
    #entry=None,
    #left_truncated=False,
    sort_by_duration=False,
    event_observed_color='red', 
    event_censored_color='blue', 
)
plt.xlabel('Days subscribed', fontsize=16);
plt.ylabel('Customer ID', fontsize=16);
plt.title('Customer subscription lifelines (random sample)', fontsize=20)
plt.tight_layout()

3 Data Preparation

df['customerID'].duplicated().any()

False

df = df.set_index('customerID')
df

	gender	SeniorCitizen	Partner	Dependents	tenure	PhoneService	MultipleLines	InternetService	OnlineSecurity	OnlineBackup	DeviceProtection	TechSupport	StreamingTV	StreamingMovies	Contract	PaperlessBilling	PaymentMethod	MonthlyCharges	TotalCharges	Churn
customerID
7590-VHVEG	Female	0	Yes	No	1	No	No phone service	DSL	No	Yes	No	No	No	No	Month-to-month	Yes	Electronic check	29.85	29.85	No
5575-GNVDE	Male	0	No	No	34	Yes	No	DSL	Yes	No	Yes	No	No	No	One year	No	Mailed check	56.95	1889.50	No
3668-QPYBK	Male	0	No	No	2	Yes	No	DSL	Yes	Yes	No	No	No	No	Month-to-month	Yes	Mailed check	53.85	108.15	Yes
7795-CFOCW	Male	0	No	No	45	No	No phone service	DSL	Yes	No	Yes	Yes	No	No	One year	No	Bank transfer (automatic)	42.30	1840.75	No
9237-HQITU	Female	0	No	No	2	Yes	No	Fiber optic	No	No	No	No	No	No	Month-to-month	Yes	Electronic check	70.70	151.65	Yes
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
6840-RESVB	Male	0	Yes	Yes	24	Yes	Yes	DSL	Yes	No	Yes	Yes	Yes	Yes	One year	Yes	Mailed check	84.80	1990.50	No
2234-XADUH	Female	0	Yes	Yes	72	Yes	Yes	Fiber optic	No	Yes	Yes	No	Yes	Yes	One year	Yes	Credit card (automatic)	103.20	7362.90	No
4801-JZAZL	Female	0	Yes	Yes	11	No	No phone service	DSL	Yes	No	No	No	No	No	Month-to-month	Yes	Electronic check	29.60	346.45	No
8361-LTMKD	Male	1	Yes	No	4	Yes	Yes	Fiber optic	No	No	No	No	No	No	Month-to-month	Yes	Mailed check	74.40	306.60	Yes
3186-AJIEK	Male	0	No	No	66	Yes	No	Fiber optic	Yes	No	Yes	Yes	Yes	Yes	Two year	Yes	Bank transfer (automatic)	105.65	6844.50	No

7043 rows × 20 columns

3.1 Event/Censor times/durations

df['tenure'] = df['tenure'].astype(float)

3.2 Status indicator

df['Churn'] = (df['Churn']=='Yes')

3.3 Numericals

df[df['TotalCharges'].isnull()]

	gender	SeniorCitizen	Partner	Dependents	tenure	PhoneService	MultipleLines	InternetService	OnlineSecurity	OnlineBackup	DeviceProtection	TechSupport	StreamingTV	StreamingMovies	Contract	PaperlessBilling	PaymentMethod	MonthlyCharges	TotalCharges	Churn
customerID
4472-LVYGI	Female	0	Yes	Yes	0.0	No	No phone service	DSL	Yes	No	Yes	Yes	Yes	No	Two year	Yes	Bank transfer (automatic)	52.55	NaN	False
3115-CZMZD	Male	0	No	Yes	0.0	Yes	No	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	Two year	No	Mailed check	20.25	NaN	False
5709-LVOEQ	Female	0	Yes	Yes	0.0	Yes	No	DSL	Yes	Yes	Yes	No	Yes	Yes	Two year	No	Mailed check	80.85	NaN	False
4367-NUYAO	Male	0	Yes	Yes	0.0	Yes	Yes	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	Two year	No	Mailed check	25.75	NaN	False
1371-DWPAZ	Female	0	Yes	Yes	0.0	No	No phone service	DSL	Yes	Yes	Yes	Yes	Yes	No	Two year	No	Credit card (automatic)	56.05	NaN	False
7644-OMVMY	Male	0	Yes	Yes	0.0	Yes	No	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	Two year	No	Mailed check	19.85	NaN	False
3213-VVOLG	Male	0	Yes	Yes	0.0	Yes	Yes	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	Two year	No	Mailed check	25.35	NaN	False
2520-SGTTA	Female	0	Yes	Yes	0.0	Yes	No	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	Two year	No	Mailed check	20.00	NaN	False
2923-ARZLG	Male	0	Yes	Yes	0.0	Yes	No	No	No internet service	No internet service	No internet service	No internet service	No internet service	No internet service	One year	Yes	Mailed check	19.70	NaN	False
4075-WKNIU	Female	0	Yes	Yes	0.0	Yes	Yes	DSL	No	Yes	Yes	Yes	Yes	No	Two year	No	Mailed check	73.35	NaN	False
2775-SEFEE	Male	0	No	Yes	0.0	Yes	Yes	DSL	Yes	Yes	No	Yes	No	No	Two year	Yes	Bank transfer (automatic)	61.90	NaN	False

df['TotalCharges'] = df['TotalCharges'].fillna(0.0)

df[df['TotalCharges'].isnull()]

	gender	SeniorCitizen	Partner	Dependents	tenure	PhoneService	MultipleLines	InternetService	OnlineSecurity	OnlineBackup	DeviceProtection	TechSupport	StreamingTV	StreamingMovies	Contract	PaperlessBilling	PaymentMethod	MonthlyCharges	TotalCharges	Churn
customerID

3.4 Categoricals

# Categoricals will be prepared later

4 Modeling

4.1 Survival Probability for the average customer

First we will use the Kaplan-Meier non-parametric model to estimate the survival probability of an average customer, i.e. not for an individual customer.

durations = df['tenure']
churns = df['Churn']
kmf = lifelines.KaplanMeierFitter()
kmf.fit(durations=durations, event_observed=churns, label='Estimate for Average Customer')

<lifelines.KaplanMeierFitter:"Estimate for Average Customer", fitted with 7043 total observations, 5174 right-censored observations>

kmf.survival_function_

	Estimate for Average Customer
timeline
0.0	1.000000
1.0	0.945961
2.0	0.927835
3.0	0.913725
4.0	0.901045
...	...
68.0	0.626858
69.0	0.620135
70.0	0.609657
71.0	0.602781
72.0	0.592790

73 rows × 1 columns

fig, ax = plt.subplots(figsize=(12,7))
kmf.plot_survival_function(ax=ax)
ax.set_title('Kaplan-Meier Customer Survival Probability - All Customers')
ax.set_xlabel('tenure (Months)')
ax.set_ylabel('Survival Probability')
plt.hlines(y=0.7, xmin=0, xmax=70, linestyles='dotted', colors=['red'])
plt.show()

As seen from the chart, the survival probability $S(t)$ declines with tenure. Customer retention efforts can be costly and a company will lose profitability if such measures are applied without discretion. Let's say we want to have our customer retention procedure only kick in when $S(t)$ falls below 70%. From the chart it appears that this will have to happen when the tenure exceeds 51 months. This model represents the average customer which means customer retention will always happen when a customer's tenure exceeds 51 months.

sum(durations==0)

11

from lifelines.plotting import qq_plot
fig, axes = plt.subplots(2, 2, figsize=(10, 10))
timeline = np.linspace(0, 0.25, 100)
wf = lifelines.WeibullFitter().fit(durations+0.001, churns, label="Weibull", timeline=timeline)
lnf = lifelines.LogNormalFitter().fit(durations+0.001, churns, label="Log Normal", timeline=timeline)
# plot what we just fit, along with the KMF estimate
kmf.plot_cumulative_density(ax=axes[0][0], ci_show=False)
wf.plot_cumulative_density(ax=axes[0][0], ci_show=False)
qq_plot(wf, ax=axes[0][1])
kmf.plot_cumulative_density(ax=axes[1][0], ci_show=False)
lnf.plot_cumulative_density(ax=axes[1][0], ci_show=False)
qq_plot(lnf, ax=axes[1][1]);

from lifelines.utils import find_best_parametric_model
best_model, best_aic_ = find_best_parametric_model(durations+0.001, churns)
best_model.plot_hazard();

4.2 Survival Probability for the average customer based on segments

The previous approach is unfocused and may lead to a lot of waste. We can improve the effectiveness somewhat by studying the survival functions for different segments of the customer population. Segmentation can be performed based on the classes in categorical predictors. Let's demonstrate this approach by making use of the 'InternetService' predictor.

#create Kaplan-Meier curve plots segmented by categoricals
def plot_categorical_survival(dft, feature, duration='tenure', event='Churn', ax=None):
    for cat in dft[feature].unique():
        idxs = dft[feature]==cat
        kmft = lifelines.KaplanMeierFitter()
        kmft.fit(dft[idxs][duration], event_observed=dft[idxs][event], label=cat)
        kmft.plot(ax=ax, label=cat)

fig_pmt, ax_pmt = plt.subplots(figsize=(12,7))
plot_categorical_survival(dft=df, feature='PaymentMethod', ax=ax_pmt)
ax_pmt.set_title('Customer Churn by Payment Method')
ax_pmt.set_xlabel('tenure (Months)')
ax_pmt.set_ylabel('Survival Probability')
plt.hlines(y=0.7, xmin=0, xmax=70, linestyles='dotted', colors=['red'])
plt.show()

fig_contract, ax_contract = plt.subplots(figsize=(12,7))
plot_categorical_survival(dft=df, feature='Contract', ax=ax_contract)
ax_contract.set_title('Customer Churn by Contract Type')
ax_contract.set_xlabel('tenure (Months)')
ax_contract.set_ylabel('Survival Probability')
plt.hlines(y=0.7, xmin=0, xmax=70, linestyles='dotted', colors=['red'])
plt.show()

fig_dep, ax_dep = plt.subplots(figsize=(12,7))
plot_categorical_survival(dft=df, feature='Dependents', ax=ax_dep)
ax_dep.set_title('Customer Churn, Dependents vs. No Dependents')
ax_dep.set_xlabel('tenure (Months)')
ax_dep.set_ylabel('Survival Probability')
plt.hlines(y=0.7, xmin=0, xmax=70, linestyles='dotted', colors=['red'])
plt.show()

fig_dep, ax_dep = plt.subplots(figsize=(12,7))
plot_categorical_survival(dft=df, feature='InternetService', ax=ax_dep)
ax_dep.set_title('Customer Churn by InternetService')
ax_dep.set_xlabel('tenure (Months)')
ax_dep.set_ylabel('Survival Probability')
plt.hlines(y=0.7, xmin=0, xmax=70, linestyles='dotted', colors=['red'])
plt.show()

From the chart we see that we can be more effective by only activating the retention procedure for fiber optic customers (when their tenure exceeds about 26 months).

4.3 Estimate Hazard

Before we can use another approach to model the average customer, we need to make use of the concept of hazard. The hazard function or hazard rate was defined above. It is the event rate in the instant after $t$, given survival past $t$. In the context of customer churn the hazard is an important indicator and needs to be as low as possible. If it goes up it indicates something has changed between the company and a customer. Such a situation calls for intervention.

On a more technical level, why should we care about the hazard function? Firstly, it is related to the survival function $S(t)$. Secondly, an important technique for modeling survival time as a function of covariates (in this case customer features) depends on the hazard function. This kind of technique will be necessary to analyze an individual customer, rather than the whole population.

4.4 Nelson-Aalen estimator

The Nelson-Aalen estimator estimates the cumulative hazard rate. Like the Kaplan-Meier estimator, it is non-parametric.

4.4.1 Cumulative hazard function

from lifelines import NelsonAalenFitter
naf = NelsonAalenFitter(alpha=0.05, nelson_aalen_smoothing=False)
naf.fit(durations=durations, event_observed=churns)
naf.cumulative_hazard_.head()

	NA_estimate
timeline
0.0	0.000000
1.0	0.054039
2.0	0.073201
3.0	0.088408
4.0	0.102286

naf.plot();

fig, ax = plt.subplots(figsize=(12,7))
var = 'InternetService'
for name, dft in df.groupby(var):
    idxs = df[var]==name
    naf = lifelines.NelsonAalenFitter().fit(dft['tenure'][idxs], dft['Churn'][idxs], alpha=0.05, label=name)
    naf.plot(ax=ax)
ax.set_title(var)
ax.set_xlabel('tenure (Months)')
ax.set_ylabel('cumulative hazard rate')
plt.show()

4.4.2 Hazard function

fig, ax = plt.subplots(figsize=(12,7))
var = 'InternetService'
for name, dft in df.groupby(var):
    idxs = df[var]==name
    naf = lifelines.NelsonAalenFitter().fit(dft['tenure'][idxs], dft['Churn'][idxs], alpha=0.05, label=name)
    naf.plot_hazard(ax=ax, bandwidth=6)
ax.set_title(var)
ax.set_xlabel('tenure (Months)')
ax.set_ylabel('hazard rate')
plt.show()

4.2 Survival Regression Model

So far, we have only looked at analyzing groups of customers. Now we are going to setup a model for an individual customer. This will increase the effectiveness of the customer retention mechanism. We next make use of the Cox Proportional Hazards model. This will allow us to model the survival time as a function of the covariates (i.e. the customer features).

4.2.1 Proportional Hazards Assumption

The proportional hazards assumption states

$h(t|x_i) = h_0(t) \text{exp} (\sum_{j=1}^{p}x_{ij}\beta_j)$

where $h_0(t)$, an unspecified function, is known as the baseline hazard. It is the hazard function for an individual with features $x_{i1} = ... = x_{ip} = 0$. The term proportional hazards derives from the fact that the hazard function for an individual with feature vector $x_i$ is some unknown function $h_0(t)$ times the factor $\text{exp} (\sum_{j=1}^{p}x_{ij}\beta_j)$. The latter quantity is called the relative risk for the feature vector $x_i = (x_{i1}, ..., x_{ip})^T$, relative to that for the feature vector $x_i = (0, ..., 0)^T$.

The baseline hazard function $h_0(t)$ can take on any form. The only assumption is that a one-unit increase in $x_{ij}$ corresponds to an increase in $h(t|x_i)$ by a factor of exp($\beta_j$).

4.2.2 Cox Proportional Hazards Regression

We need to do some pre-processing before applying this model.

df

	gender	SeniorCitizen	Partner	Dependents	tenure	PhoneService	MultipleLines	InternetService	OnlineSecurity	OnlineBackup	DeviceProtection	TechSupport	StreamingTV	StreamingMovies	Contract	PaperlessBilling	PaymentMethod	MonthlyCharges	TotalCharges	Churn
customerID
7590-VHVEG	Female	0	Yes	No	1.0	No	No phone service	DSL	No	Yes	No	No	No	No	Month-to-month	Yes	Electronic check	29.85	29.85	False
5575-GNVDE	Male	0	No	No	34.0	Yes	No	DSL	Yes	No	Yes	No	No	No	One year	No	Mailed check	56.95	1889.50	False
3668-QPYBK	Male	0	No	No	2.0	Yes	No	DSL	Yes	Yes	No	No	No	No	Month-to-month	Yes	Mailed check	53.85	108.15	True
7795-CFOCW	Male	0	No	No	45.0	No	No phone service	DSL	Yes	No	Yes	Yes	No	No	One year	No	Bank transfer (automatic)	42.30	1840.75	False
9237-HQITU	Female	0	No	No	2.0	Yes	No	Fiber optic	No	No	No	No	No	No	Month-to-month	Yes	Electronic check	70.70	151.65	True
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
6840-RESVB	Male	0	Yes	Yes	24.0	Yes	Yes	DSL	Yes	No	Yes	Yes	Yes	Yes	One year	Yes	Mailed check	84.80	1990.50	False
2234-XADUH	Female	0	Yes	Yes	72.0	Yes	Yes	Fiber optic	No	Yes	Yes	No	Yes	Yes	One year	Yes	Credit card (automatic)	103.20	7362.90	False
4801-JZAZL	Female	0	Yes	Yes	11.0	No	No phone service	DSL	Yes	No	No	No	No	No	Month-to-month	Yes	Electronic check	29.60	346.45	False
8361-LTMKD	Male	1	Yes	No	4.0	Yes	Yes	Fiber optic	No	No	No	No	No	No	Month-to-month	Yes	Mailed check	74.40	306.60	True
3186-AJIEK	Male	0	No	No	66.0	Yes	No	Fiber optic	Yes	No	Yes	Yes	Yes	Yes	Two year	Yes	Bank transfer (automatic)	105.65	6844.50	False

7043 rows × 20 columns

X = pd.get_dummies(df.drop(['Churn', 'tenure'], axis=1), drop_first=True)
y = df.loc[:, ['tenure', 'Churn']]
df_clean = pd.concat([X, y], axis=1)

df_clean

	SeniorCitizen	MonthlyCharges	TotalCharges	gender_Male	Partner_Yes	Dependents_Yes	PhoneService_Yes	MultipleLines_No phone service	MultipleLines_Yes	InternetService_Fiber optic	InternetService_No	OnlineSecurity_No internet service	OnlineSecurity_Yes	OnlineBackup_No internet service	OnlineBackup_Yes	DeviceProtection_No internet service	DeviceProtection_Yes	TechSupport_No internet service	TechSupport_Yes	StreamingTV_No internet service	StreamingTV_Yes	StreamingMovies_No internet service	StreamingMovies_Yes	Contract_One year	Contract_Two year	PaperlessBilling_Yes	PaymentMethod_Credit card (automatic)	PaymentMethod_Electronic check	PaymentMethod_Mailed check	tenure	Churn
customerID
7590-VHVEG	0	29.85	29.85	0	1	0	0	1	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	1	0	1	0	1.0	False
5575-GNVDE	0	56.95	1889.50	1	0	0	1	0	0	0	0	0	1	0	0	0	1	0	0	0	0	0	0	1	0	0	0	0	1	34.0	False
3668-QPYBK	0	53.85	108.15	1	0	0	1	0	0	0	0	0	1	0	1	0	0	0	0	0	0	0	0	0	0	1	0	0	1	2.0	True
7795-CFOCW	0	42.30	1840.75	1	0	0	0	1	0	0	0	0	1	0	0	0	1	0	1	0	0	0	0	1	0	0	0	0	0	45.0	False
9237-HQITU	0	70.70	151.65	0	0	0	1	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	1	0	2.0	True
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
6840-RESVB	0	84.80	1990.50	1	1	1	1	0	1	0	0	0	1	0	0	0	1	0	1	0	1	0	1	1	0	1	0	0	1	24.0	False
2234-XADUH	0	103.20	7362.90	0	1	1	1	0	1	1	0	0	0	0	1	0	1	0	0	0	1	0	1	1	0	1	1	0	0	72.0	False
4801-JZAZL	0	29.60	346.45	0	1	1	0	1	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	1	0	1	0	11.0	False
8361-LTMKD	1	74.40	306.60	1	1	0	1	0	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	1	4.0	True
3186-AJIEK	0	105.65	6844.50	1	0	0	1	0	0	1	0	0	1	0	0	0	1	0	1	0	1	0	1	0	1	1	0	0	0	66.0	False

7043 rows × 31 columns

df_clean.columns

Index(['SeniorCitizen', 'MonthlyCharges', 'TotalCharges', 'gender_Male',
       'Partner_Yes', 'Dependents_Yes', 'PhoneService_Yes',
       'MultipleLines_No phone service', 'MultipleLines_Yes',
       'InternetService_Fiber optic', 'InternetService_No',
       'OnlineSecurity_No internet service', 'OnlineSecurity_Yes',
       'OnlineBackup_No internet service', 'OnlineBackup_Yes',
       'DeviceProtection_No internet service', 'DeviceProtection_Yes',
       'TechSupport_No internet service', 'TechSupport_Yes',
       'StreamingTV_No internet service', 'StreamingTV_Yes',
       'StreamingMovies_No internet service', 'StreamingMovies_Yes',
       'Contract_One year', 'Contract_Two year', 'PaperlessBilling_Yes',
       'PaymentMethod_Credit card (automatic)',
       'PaymentMethod_Electronic check', 'PaymentMethod_Mailed check',
       'tenure', 'Churn'],
      dtype='object')

4.2.3 Split the data

import numpy as np
num_customers = df_clean.shape[0]

num_train = int(num_customers * 0.7)
train_customers = np.random.choice(df_clean.index, size=num_train, replace=False)
df_train = df_clean.loc[train_customers]

num_valid = int((num_customers - num_train) * 0.5)
valid_customers = np.random.choice(df_clean.drop(train_customers).index, size=num_valid, replace=False)
df_valid = df_clean.loc[valid_customers]

test_customers = df.drop(train_customers).drop(valid_customers).index
df_test = df_clean.loc[test_customers]

total = df_train.shape[0] + df_valid.shape[0] + df_test.shape[0]
print('Train', df_train.shape, df_train.shape[0]/total*100)
print('Dev', df_valid.shape, df_valid.shape[0]/total*100)
print('Test', df_test.shape, df_test.shape[0]/total*100)
print('Total:', total)
assert total == df.shape[0]

Train (4930, 31) 69.99858015050404
Dev (1056, 31) 14.99361067726821
Test (1057, 31) 15.007809172227743
Total: 7043

cph = lifelines.CoxPHFitter()
cph.fit(df_train, duration_col='tenure', event_col='Churn', show_progress=False)
cph.print_summary()

---------------------------------------------------------------------------
LinAlgError                               Traceback (most recent call last)
/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in _newton_rhapson_for_efron_model(self, X, T, E, weights, entries, initial_point, step_size, precision, show_progress, max_steps)
   1523             try:
-> 1524                 inv_h_dot_g_T = spsolve(-h, g, assume_a="pos", check_finite=False)
   1525             except (ValueError, LinAlgError) as e:

/usr/local/lib/python3.7/dist-packages/scipy/linalg/basic.py in solve(a, b, sym_pos, lower, overwrite_a, overwrite_b, debug, check_finite, assume_a, transposed)
    252                            overwrite_b=overwrite_b)
--> 253         _solve_check(n, info)
    254         rcond, info = pocon(lu, anorm)

/usr/local/lib/python3.7/dist-packages/scipy/linalg/basic.py in _solve_check(n, info, lamch, rcond)
     28     elif 0 < info:
---> 29         raise LinAlgError('Matrix is singular.')
     30 

LinAlgError: Matrix is singular.

During handling of the above exception, another exception occurred:

ConvergenceError                          Traceback (most recent call last)
<ipython-input-80-a2b2a571ba39> in <module>()
      1 cph = lifelines.CoxPHFitter()
----> 2 cph.fit(df_train, duration_col='tenure', event_col='Churn', show_progress=False)
      3 cph.print_summary()

/usr/local/lib/python3.7/dist-packages/lifelines/utils/__init__.py in f(model, *args, **kwargs)
     54         def f(model, *args, **kwargs):
     55             cls.set_censoring_type(model, cls.RIGHT)
---> 56             return function(model, *args, **kwargs)
     57 
     58         return f

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in fit(self, df, duration_col, event_col, show_progress, initial_point, strata, step_size, weights_col, cluster_col, robust, batch_mode, timeline, formula, entry_col)
    302             timeline=timeline,
    303             formula=formula,
--> 304             entry_col=entry_col,
    305         )
    306         return self

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in _fit_model(self, *args, **kwargs)
    614     def _fit_model(self, *args, **kwargs):
    615         if self.baseline_estimation_method == "breslow":
--> 616             return self._fit_model_breslow(*args, **kwargs)
    617         elif self.baseline_estimation_method == "spline":
    618             return self._fit_model_spline(*args, **kwargs)

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in _fit_model_breslow(self, *args, **kwargs)
    627         )
    628         if utils.CensoringType.is_right_censoring(self):
--> 629             model.fit(*args, **kwargs)
    630             return model
    631         else:

/usr/local/lib/python3.7/dist-packages/lifelines/utils/__init__.py in f(model, *args, **kwargs)
     54         def f(model, *args, **kwargs):
     55             cls.set_censoring_type(model, cls.RIGHT)
---> 56             return function(model, *args, **kwargs)
     57 
     58         return f

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in fit(self, df, duration_col, event_col, show_progress, initial_point, strata, step_size, weights_col, cluster_col, robust, batch_mode, timeline, formula, entry_col)
   1263             initial_point=initial_point,
   1264             show_progress=show_progress,
-> 1265             step_size=step_size,
   1266         )
   1267 

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in _fit_model(self, X, T, E, weights, entries, initial_point, step_size, show_progress)
   1384     ):
   1385         beta_, ll_, hessian_ = self._newton_rhapson_for_efron_model(
-> 1386             X, T, E, weights, entries, initial_point=initial_point, step_size=step_size, show_progress=show_progress
   1387         )
   1388 

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in _newton_rhapson_for_efron_model(self, X, T, E, weights, entries, initial_point, step_size, precision, show_progress, max_steps)
   1535                             CONVERGENCE_DOCS
   1536                         ),
-> 1537                         e,
   1538                     )
   1539                 else:

ConvergenceError: Convergence halted due to matrix inversion problems. Suspicion is high collinearity. Please see the following tips in the lifelines documentation: https://lifelines.readthedocs.io/en/latest/Examples.html#problems-with-convergence-in-the-cox-proportional-hazard-modelMatrix is singular.

# Suspicion is high collinearity. Please see the following tips in the lifelines documentation: 
# https://lifelines.readthedocs.io/en/latest/Examples.html#problems-with-convergence-in-the-cox-proportional-hazard-modelMatrix is singular.

To address this problem we will make use of the variance inflation factor (VIF) approach.

from statsmodels.stats.outliers_influence import variance_inflation_factor

def calc_vif(X):
    vif = pd.DataFrame()
    vif["variables"] = X.columns
    vif["VIF"] = [variance_inflation_factor(X.values, i) for i in range(X.shape[1])]
    return(vif)

feature_vif = calc_vif(df_train.drop(['Churn', 'tenure'], axis=1))
feature_vif.sort_values('VIF')

	variables	VIF
3	gender_Male	1.002632
0	SeniorCitizen	1.155752
25	PaperlessBilling_Yes	1.219196
5	Dependents_Yes	1.370769
4	Partner_Yes	1.430055
23	Contract_One year	1.537983
26	PaymentMethod_Credit card (automatic)	1.576695
28	PaymentMethod_Mailed check	1.799718
27	PaymentMethod_Electronic check	1.976341
24	Contract_Two year	2.266687
2	TotalCharges	3.986928
12	OnlineSecurity_Yes	6.299991
18	TechSupport_Yes	6.460155
14	OnlineBackup_Yes	6.771743
16	DeviceProtection_Yes	6.921651
8	MultipleLines_Yes	7.275559
20	StreamingTV_Yes	23.770145
22	StreamingMovies_Yes	24.167640
7	MultipleLines_No phone service	61.569033
9	InternetService_Fiber optic	148.414936
1	MonthlyCharges	846.284329
6	PhoneService_Yes	1743.106075
11	OnlineSecurity_No internet service	inf
15	DeviceProtection_No internet service	inf
17	TechSupport_No internet service	inf
19	StreamingTV_No internet service	inf
21	StreamingMovies_No internet service	inf
10	InternetService_No	inf
13	OnlineBackup_No internet service	inf

features_keep = feature_vif.loc[feature_vif['VIF'] < 10, 'variables'].tolist()
features_keep.extend(['tenure', 'Churn'])
features_keep

['SeniorCitizen',
 'TotalCharges',
 'gender_Male',
 'Partner_Yes',
 'Dependents_Yes',
 'MultipleLines_Yes',
 'OnlineSecurity_Yes',
 'OnlineBackup_Yes',
 'DeviceProtection_Yes',
 'TechSupport_Yes',
 'Contract_One year',
 'Contract_Two year',
 'PaperlessBilling_Yes',
 'PaymentMethod_Credit card (automatic)',
 'PaymentMethod_Electronic check',
 'PaymentMethod_Mailed check',
 'tenure',
 'Churn']

cph = lifelines.CoxPHFitter()
cph.fit(
    df_train.loc[:, features_keep], 
    duration_col='tenure', 
    event_col='Churn', 
    show_progress=False)
cph.print_summary()

model	lifelines.CoxPHFitter
duration col	'tenure'
event col	'Churn'
baseline estimation	breslow
number of observations	4930
number of events observed	1291
partial log-likelihood	-8752.37
time fit was run	2022-04-11 17:48:43 UTC

	coef	exp(coef)	se(coef)	coef lower 95%	coef upper 95%	exp(coef) lower 95%	exp(coef) upper 95%	cmp to	z	p	-log2(p)
SeniorCitizen	0.17	1.19	0.07	0.04	0.31	1.04	1.36	0.00	2.51	0.01	6.37
TotalCharges	-0.00	1.00	0.00	-0.00	-0.00	1.00	1.00	0.00	-26.70	<0.005	519.39
gender_Male	-0.00	1.00	0.06	-0.11	0.11	0.89	1.11	0.00	-0.05	0.96	0.06
Partner_Yes	-0.21	0.81	0.07	-0.34	-0.08	0.71	0.92	0.00	-3.16	<0.005	9.30
Dependents_Yes	-0.25	0.78	0.08	-0.42	-0.09	0.66	0.91	0.00	-3.04	<0.005	8.73
MultipleLines_Yes	0.51	1.67	0.06	0.39	0.63	1.48	1.88	0.00	8.29	<0.005	53.03
OnlineSecurity_Yes	-0.18	0.84	0.08	-0.33	-0.03	0.72	0.97	0.00	-2.29	0.02	5.50
OnlineBackup_Yes	0.04	1.04	0.07	-0.10	0.17	0.91	1.18	0.00	0.54	0.59	0.76
DeviceProtection_Yes	0.33	1.39	0.07	0.20	0.46	1.22	1.58	0.00	5.04	<0.005	21.03
TechSupport_Yes	0.01	1.01	0.08	-0.15	0.16	0.86	1.17	0.00	0.08	0.94	0.09
Contract_One year	-1.95	0.14	0.11	-2.17	-1.73	0.11	0.18	0.00	-17.53	<0.005	226.23
Contract_Two year	-4.43	0.01	0.20	-4.83	-4.02	0.01	0.02	0.00	-21.59	<0.005	340.97
PaperlessBilling_Yes	0.49	1.63	0.07	0.35	0.62	1.43	1.86	0.00	7.22	<0.005	40.83
PaymentMethod_Credit card (automatic)	-0.11	0.89	0.11	-0.33	0.10	0.72	1.10	0.00	-1.05	0.29	1.76
PaymentMethod_Electronic check	0.58	1.79	0.09	0.41	0.75	1.51	2.12	0.00	6.67	<0.005	35.14
PaymentMethod_Mailed check	0.11	1.12	0.11	-0.09	0.32	0.91	1.38	0.00	1.08	0.28	1.84

Concordance	0.92
Partial AIC	17536.74
log-likelihood ratio test	3204.21 on 16 df
-log2(p) of ll-ratio test	inf

The summary provide some key information:

• p-values indicate how significant a feature is
• exp(coef) indicate by how much the hazard will increase when the feature is increased by 1 unit
• concordance

5 Evaluation

5.1 Plot the coefficients

fig_coef, ax_coef = plt.subplots(figsize=(12,10))
ax_coef.set_title('Survival Regression: Coefficients and Confident Intervals')
cph.plot(ax=ax_coef);

cph.plot_partial_effects_on_outcome(
    covariates='Contract_Two year', 
    values=[0, 1], 
    plot_baseline=True,
    cmap='coolwarm');

cph.plot_partial_effects_on_outcome(
    covariates='Contract_One year', 
    values=[0, 1], 
    plot_baseline=True,
    cmap='coolwarm');

5.2 Predict on the train set

surv_pred_train = cph.predict_survival_function(df_train)
surv_median_pred_train = cph.predict_median(df_train)

surv_pred_train

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0.952700	0.994119	0.992714	3.242608e-26	2.965170e-31	6.161052e-42	1.154639e-17	0.998442	6.059762e-14	4.038744e-76	0.442016	0.997464	6.271862e-21	1.330202e-54	0.952425	0.988122	0.001093	0.901577	4.395987e-45	0.958340	0.048020	4.949130e-36	0.856213	1.805022e-26	0.065927	0.138829	0.988638	0.766530	6.754573e-32	8.330993e-93	0.135549	0.000005	2.383791e-30	5.191583e-07	0.680388	1.540378e-33	0.001630	0.064451	0.007276	6.021421e-34	0.053977	6.897606e-29	0.996619	0.002896	0.868036	4.248972e-116	4.082914e-56	8.874529e-10	2.295540e-103	1.751954e-36	0.987658	5.567161e-54	6.661385e-42	0.000602	0.001548	1.040810e-52	3.792950e-27	0.949734	0.000010	5.750660e-80	3.309653e-40	0.782493	0.946038	3.119315e-72	0.000191	0.021644	0.996548	0.063351	0.994105	5.581321e-53	3.265667e-09	0.755672	0.800418	6.475050e-09	0.987916	4.654274e-31	0.800108	1.967644e-52	5.347392e-51	0.996525	0.004390	5.761926e-18	0.994869	0.995563	1.718653e-17	0.863828	2.121742e-37	1.115346e-149	3.400960e-37	2.057723e-71	2.011644e-24	5.899554e-59	0.988415	1.065382e-23	1.137182e-58	1.284314e-42	5.965927e-10	4.508654e-12	4.589287e-51	0.921989	0.001732	0.976832	0.984248	2.054707e-20	0.999146	0.868015	0.476195	1.101218e-15	0.444543	0.995327	0.965074	0.997806	0.021858	6.085875e-72	6.348532e-35	0.053182	0.000499	1.463056e-121	4.846752e-27	4.851220e-53	2.355018e-72	1.436628e-56	9.836535e-09	0.937940	8.393050e-13	2.254560e-16	0.000074	2.064702e-30	9.578809e-07	1.567584e-58	5.676123e-22	0.986061	2.169495e-56	0.727675	1.899959e-12	3.520543e-59	0.646214	1.271644e-23	0.357500	3.269778e-44	3.588862e-27	0.008533	0.388963	1.166133e-36	0.042829	0.597068	1.970068e-37	0.890809	1.439050e-124	0.977613	0.583156	0.037370	1.784539e-90	0.823855	9.026587e-33	0.000017	0.606875	1.932946e-37	0.995476	0.000118	5.289869e-108	9.965886e-20	1.769007e-56	9.272172e-23	0.013242	0.000004	6.841820e-18	0.988931	0.981397	0.265799	0.999136	3.189160e-19	4.891637e-67	1.547319e-17	7.564053e-65	1.446101e-10	1.747081e-144	2.252017e-35	2.054236e-37	0.993009	3.843175e-21	0.024761	5.291163e-21	0.789228	3.362535e-128	0.030370	0.618653	0.433443	3.083469e-37	4.308575e-08	1.383298e-57	3.063570e-33	0.993339	4.884774e-28	0.650592	0.591169	0.413200	0.000085	0.997957	0.090941	0.793223	0.017518	6.478952e-30	2.463881e-12	0.997133	4.480593e-205	0.322236	6.707908e-61	0.782226	0.992074	0.348105	0.951148	0.575848	0.351050	0.720965	0.000007	5.433747e-58	0.142159	0.794014	0.988662	0.659283	0.000002	0.407896	2.793589e-105	1.374805e-39	3.123138e-07	0.000201	0.373509	1.127523e-40	3.596035e-61	0.069807	5.888406e-109	6.290640e-42	2.438024e-39	0.010247	7.388500e-21	0.009903	0.951228	4.014612e-26	0.989604	2.943344e-105	9.000143e-97	3.161826e-07	4.223266e-92	0.994634	0.609296	0.210100	0.224179	4.914084e-75	0.882822	0.601228	0.818496	0.223296	1.792826e-41	0.985791	0.623684	0.000372	0.983341	0.991270	0.792664	0.001164	2.849683e-59	0.935085	0.000646	0.975335	1.124160e-07	9.500506e-63	0.980284	5.425446e-27	0.957142	3.654063e-37	1.818059e-48	0.989271	0.802639	6.765562e-54	2.695495e-32	7.226065e-10	2.541642e-26	0.421769	0.033139	0.014716	0.010991	2.213238e-38	1.174703e-71	2.519446e-91	2.517744e-23	1.122518e-28	2.413362e-16	0.653761	0.959369	0.790252	0.998229	1.299603e-91	3.419521e-79	1.138427e-72	0.992592	0.000576	0.651066	3.801880e-40	3.858562e-44	0.580213	0.984303	9.354549e-10	9.588656e-17	0.726821	0.871751	9.016146e-57	2.546208e-24	0.847553	0.628071	2.073989e-49	3.026186e-47	0.681042	0.977577	3.566286e-59	5.715706e-90	0.003244	2.002151e-11	6.063390e-57	0.753276	0.194299	1.288556e-63	0.773883	0.000014	0.981121	1.920563e-17	2.080010e-07	0.307784	0.958194	4.282976e-28

73 rows × 4930 columns

kmf.survival_function_

	Estimate for Average Customer
timeline
0.0	1.000000
1.0	0.945961
2.0	0.927835
3.0	0.913725
4.0	0.901045
...	...
68.0	0.626858
69.0	0.620135
70.0	0.609657
71.0	0.602781
72.0	0.592790

73 rows × 1 columns

import plotly.graph_objs as go
fig = go.Figure()
buttons = []
fig.add_trace(
    go.Scatter(
        x=kmf.survival_function_.index, 
        #.y=kmf.survival_function_['KM_estimate'],
        y=kmf.survival_function_,
        marker_color="black",
        #.name='KM estimate'
    )
)
buttons.append(dict(method='restyle', 
                    #.label='KM Estimate', 
                    #.args=[{'y': [kmf.survival_function_['KM_estimate'].values]}]))
                    args=[{'y': [kmf.survival_function_.values]}]))

for customerID in surv_pred_train.columns[0:50]:
    #.color = 'red' if df_train_clean.loc[customerID, 'status'] else 'blue'
    color = 'red' if df_train.loc[customerID, 'Churn'] else 'blue'
    customer_probabilities = surv_pred_train[customerID]
    fig.add_trace(
        go.Scatter(
            x=surv_pred_train.index, 
            y=customer_probabilities,
            marker_color=color,
            name=customerID
        )
    )
    buttons.append(dict(method='restyle',
                        label=customerID,
                        args=[{'y': [customer_probabilities.values]}])
                  )
updatemenu = [{}]
updatemenu[0]['buttons'] = buttons
updatemenu[0]['direction'] = 'down'
updatemenu[0]['showactive'] = True

# update layout and show figure
fig.update_layout(updatemenus=updatemenu)

surv_pred_train.columns[0:5]

Index(['7923-IYJWY', '3665-JATSN', '3214-IYUUQ', '5181-OABFK', '0689-DSXGL'], dtype='object')

surv_median_pred_train

7923-IYJWY     inf
3665-JATSN    13.0
3214-IYUUQ     inf
5181-OABFK     inf
0689-DSXGL    70.0
              ... 
7503-QQRVF    31.0
9971-ZWPBF    46.0
8612-GXIDD    70.0
8065-QBYTO     inf
7764-BDPEE    23.0
Name: 0.5, Length: 4930, dtype: float64

patient_name = surv_pred_train.columns[19]
pred_median = surv_median_pred_train
ax = surv_pred_train[patient_name].plot()
kmf.plot(ax=ax, label='KM Estimator')
plt.vlines(df_train.loc[patient_name, 'tenure'], ymin=0, ymax=1)
plt.vlines(surv_median_pred_train[patient_name], ymin=0, ymax=1, color='red')

<matplotlib.collections.LineCollection at 0x7fa310cd6ed0>

5.3 Predict on the validation set

surv_pred_valid = cph.predict_survival_function(df_valid)
surv_median_pred_valid = cph.predict_median(df_valid)

patient_name = surv_pred_valid.columns[100]
pred_median = surv_median_pred_valid
ax = surv_pred_valid[patient_name].plot()
kmf.plot(ax=ax, label='KM Estimator')
plt.vlines(df_valid.loc[patient_name, 'tenure'], ymin=0, ymax=1)
plt.vlines(surv_median_pred_valid[patient_name], ymin=0, ymax=1, color='red');

ax = surv_pred_valid.iloc[:, 0:6].plot();
kmf.plot(ax=ax, label='KM Estimator');

5.4 Concordance

https://stats.stackexchange.com/questions/478294/how-the-concordance-index-is-calculated-in-cox-model-if-the-actual-event-times-a/478305#478305

from lifelines.utils import concordance_index
concordance_index(df_valid['tenure'], -cph.predict_partial_hazard(df_valid), df_valid['Churn'])

0.9264000512153804

5.5 Check assumptions

%%time
cph.check_assumptions(df_train.loc[:, features_keep], p_value_threshold=0.05, show_plots=True)

The ``p_value_threshold`` is set at 0.05. Even under the null hypothesis of no violations, some
covariates will be below the threshold by chance. This is compounded when there are many covariates.
Similarly, when there are lots of observations, even minor deviances from the proportional hazard
assumption will be flagged.

With that in mind, it's best to use a combination of statistical tests and visual tests to determine
the most serious violations. Produce visual plots using ``check_assumptions(..., show_plots=True)``
and looking for non-constant lines. See link [A] below for a full example.

null_distribution	chi squared
degrees_of_freedom	1
model	<lifelines.CoxPHFitter: fitted with 4930 total...
test_name	proportional_hazard_test

		test_statistic	p	-log2(p)
Contract_One year	km	59.60	<0.005	46.29
	rank	44.75	<0.005	35.38
Contract_Two year	km	53.45	<0.005	41.78
	rank	36.86	<0.005	29.55
Dependents_Yes	km	0.36	0.55	0.87
	rank	0.02	0.89	0.16
DeviceProtection_Yes	km	0.42	0.51	0.96
	rank	1.64	0.20	2.32
MultipleLines_Yes	km	4.20	0.04	4.63
	rank	14.12	<0.005	12.51
OnlineBackup_Yes	km	0.14	0.71	0.50
	rank	0.02	0.88	0.19
OnlineSecurity_Yes	km	3.64	0.06	4.15
	rank	4.69	0.03	5.04
PaperlessBilling_Yes	km	0.08	0.78	0.36
	rank	0.48	0.49	1.04
Partner_Yes	km	0.04	0.85	0.23
	rank	0.46	0.50	1.00
PaymentMethod_Credit card (automatic)	km	6.36	0.01	6.42
	rank	4.97	0.03	5.27
PaymentMethod_Electronic check	km	4.88	0.03	5.20
	rank	2.27	0.13	2.92
PaymentMethod_Mailed check	km	3.41	0.06	3.95
	rank	0.52	0.47	1.09
SeniorCitizen	km	8.42	<0.005	8.07
	rank	4.85	0.03	5.18
TechSupport_Yes	km	4.43	0.04	4.82
	rank	7.02	0.01	6.95
TotalCharges	km	623.93	<0.005	455.04
	rank	385.43	<0.005	282.65
gender_Male	km	1.52	0.22	2.20
	rank	0.47	0.49	1.02

1. Variable 'SeniorCitizen' failed the non-proportional test: p-value is 0.0037.

   Advice: with so few unique values (only 2), you can include `strata=['SeniorCitizen', ...]` in
the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


2. Variable 'TotalCharges' failed the non-proportional test: p-value is <5e-05.

   Advice 1: the functional form of the variable 'TotalCharges' might be incorrect. That is, there
may be non-linear terms missing. The proportional hazard test used is very sensitive to incorrect
functional forms. See documentation in link [D] below on how to specify a functional form.

   Advice 2: try binning the variable 'TotalCharges' using pd.cut, and then specify it in
`strata=['TotalCharges', ...]` in the call in `.fit`. See documentation in link [B] below.

   Advice 3: try adding an interaction term with your time variable. See documentation in link [C]
below.


   Bootstrapping lowess lines. May take a moment...


3. Variable 'MultipleLines_Yes' failed the non-proportional test: p-value is 0.0002.

   Advice: with so few unique values (only 2), you can include `strata=['MultipleLines_Yes', ...]`
in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


4. Variable 'OnlineSecurity_Yes' failed the non-proportional test: p-value is 0.0303.

   Advice: with so few unique values (only 2), you can include `strata=['OnlineSecurity_Yes', ...]`
in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


5. Variable 'TechSupport_Yes' failed the non-proportional test: p-value is 0.0081.

   Advice: with so few unique values (only 2), you can include `strata=['TechSupport_Yes', ...]` in
the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


6. Variable 'Contract_One year' failed the non-proportional test: p-value is <5e-05.

   Advice: with so few unique values (only 2), you can include `strata=['Contract_One year', ...]`
in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


7. Variable 'Contract_Two year' failed the non-proportional test: p-value is <5e-05.

   Advice: with so few unique values (only 2), you can include `strata=['Contract_Two year', ...]`
in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


8. Variable 'PaymentMethod_Credit card (automatic)' failed the non-proportional test: p-value is 0.0117.

   Advice: with so few unique values (only 2), you can include `strata=['PaymentMethod_Credit card
(automatic)', ...]` in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


9. Variable 'PaymentMethod_Electronic check' failed the non-proportional test: p-value is 0.0271.

   Advice: with so few unique values (only 2), you can include `strata=['PaymentMethod_Electronic
check', ...]` in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


---
[A]  https://lifelines.readthedocs.io/en/latest/jupyter_notebooks/Proportional%20hazard%20assumption.html
[B]  https://lifelines.readthedocs.io/en/latest/jupyter_notebooks/Proportional%20hazard%20assumption.html#Bin-variable-and-stratify-on-it
[C]  https://lifelines.readthedocs.io/en/latest/jupyter_notebooks/Proportional%20hazard%20assumption.html#Introduce-time-varying-covariates
[D]  https://lifelines.readthedocs.io/en/latest/jupyter_notebooks/Proportional%20hazard%20assumption.html#Modify-the-functional-form
[E]  https://lifelines.readthedocs.io/en/latest/jupyter_notebooks/Proportional%20hazard%20assumption.html#Stratification

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assumptions_results = lifelines.statistics.proportional_hazard_test(
    cph, df_train.loc[:, features_keep], time_transform='rank'
)
assumptions_results.print_summary()

time_transform	rank
null_distribution	chi squared
degrees_of_freedom	1
model	<lifelines.CoxPHFitter: fitted with 4930 total...
test_name	proportional_hazard_test

	test_statistic	p	-log2(p)
Contract_One year	44.75	<0.005	35.38
Contract_Two year	36.86	<0.005	29.55
Dependents_Yes	0.02	0.89	0.16
DeviceProtection_Yes	1.64	0.20	2.32
MultipleLines_Yes	14.12	<0.005	12.51
OnlineBackup_Yes	0.02	0.88	0.19
OnlineSecurity_Yes	4.69	0.03	5.04
PaperlessBilling_Yes	0.48	0.49	1.04
Partner_Yes	0.46	0.50	1.00
PaymentMethod_Credit card (automatic)	4.97	0.03	5.27
PaymentMethod_Electronic check	2.27	0.13	2.92
PaymentMethod_Mailed check	0.52	0.47	1.09
SeniorCitizen	4.85	0.03	5.18
TechSupport_Yes	7.02	0.01	6.95
TotalCharges	385.43	<0.005	282.65
gender_Male	0.47	0.49	1.02